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Knots and Other New Topological Effects in Liquid Crystals and Colloids

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Knots and Other New Topological Effects in Liquid Crystals and Colloids Review: Knots and other new topological effects in liquid crystals and colloids Ivan I. Smalyukh1,2* 1Department of Physics, Department of Electrical, Computer and Energy Engineering, Materials Science and Engineering Program and Soft Materials Research Center, University of Colorado, Boulder, CO 80309, USA 2Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, CO 80309, USA *Email: [email protected] Abstract: Humankind has been obsessed with knots in religion, culture and daily life for millennia while physicists like Gauss, Kelvin and Maxwell involved them in models already centuries ago. Nowadays, colloidal particle can be fabricated to have shapes of knots and links with arbitrary complexity. In liquid crystals, closed loops of singular vortex lines can be knotted by using colloidal particles and laser tweezers, as well as by confining nematic fluids into micrometer-sized droplets with complex topology. Knotted and linked colloidal particles induce knots and links of singular defects, which can be inter-linked (or not) with colloidal particle knots, revealing diversity of interactions between topologies of knotted fields and topologically nontrivial surfaces of colloidal objects. Even more diverse knotted structures emerge in nonsingular molecular alignment and magnetization fields in liquid crystals and colloidal ferromagnets. The topological solitons include hopfions, skyrmions, heliknotons, torons and other spatially localized continuous structures, which are classified based on homotopy theory, characterized by integer-valued topological invariants and often contain knotted or linked preimages, nonsingular regions of space corresponding to single points of the order parameter space. A zoo of topological solitons in liquid crystals, colloids and ferromagnets promises new breeds of information displays and a plethora of data storage, electro-optic and photonic applications. Their particle-like collective dynamics echoes coherent motions in active matter, ranging from crowds of people to schools of fish. This review discusses the state of the art in the field, as well as highlights recent developments and open questions in physics of knotted soft matter. We systematically overview knotted field configurations, the allowed transformations between them, their physical stability, and how one can use one form of knotted fields to model, create and imprint other forms. The large variety of symmetries accessible to liquid crystals and colloids offer insights on stability, transformation and emergent dynamics of fully nonsingular and singular knotted fields of fundamental and applied importance. The common thread of this review is the ability to experimentally visualize these knots in real space. The review concludes with a discussion of how the studies of knots in liquid crystals and colloids can offer insights into topologically related structures in other branches of physics, with answers to many open questions, as well as how these experimentally observable knots hold a strong potential for providing new inspirations to the mathematical knot theory. 1 Contents 1. Introduction____________________________________________________________2 2. Historic remarks_________________________________________________________5 3. Mathematical foundations_________________________________________________6 3.1. Diversity of knots and links_____________________________________________6 3.2. Homotopy theory of topological solitons and singular defects_________________8 4. Topology of nematic colloids and drops______________________________________11 4.1. Spheres and handlebodies as colloidal particles and confinement surfaces ______11 4.2. Knots as colloidal particles_____________________________________________14 4.3. Linked composite colloids______________________________________________17 4.4. Knotted vortices in nematic drops_______________________________________18 4.5. Surfaces with boundary & surface-bound defects __________________________21 5. Topological solitons in liquid crystals and colloids ______________________________22 5.1. Two-dimensional skyrmions____________________________________________22 5.2. Torons with skyrmions and knots within them______________________________23 5.3. Hopfions in ferromagnetic colloidal fluids_________________________________25 5.4. Hopfions in nonpolar liquid crystals______________________________________27 5.5. Hybrid torons & twistions______________________________________________28 5.6. Topological inter-transformations of solitons______________________________29 5.7. Heliknotons and crystals of knots _______________________________________30 6. Topological, solitonic & knotted active matter_________________________________30 6.1. Solitons as active particles in passive LCs__________________________________30 6.2. Out-of-equilibrium elastic interactions and schooling of skyrmions_____________32 6.3. Crystals of moving torons______________________________________________33 6.4. Utility of activated solitonic matter______________________________________33 7. Open questions, opportunities and perspectives _______________________________34 Acknowledgments_____________________________________________________________36 References___________________________________________________________________37 1. Introduction Topological concepts are currently at the research frontier of modern condensed matter physics, with the exciting recent developments promising to revolutionize the future of many technologies, ranging from quantum computing to pre-engineered mechanics of materials [1,2]. Although topological knot-related ideas are found in early works by Gauss, Kelvin, Tait and Maxwell [3-7], it is only recently that the concepts of topology are successful in explaining entirely new types of physical behavior of condensed matter systems [1,2], including phenomena that cannot be interpreted otherwise. Various types of knots are studied in practically all fields of physics [1-11]. The mathematical knot theory, once inspired by early models in physics [3], has become a major branch of topology with connections to statistical mechanics, models of exotic states in Bose-Einstein condensates, theories in elementary particle and nuclear physics, quantum field theory, quantum computing, solid-state physics, and many other exciting frontiers of physics research [8-11]. The knots and links in these theories are beautiful mathematical constructs that, however, typically do not manifest themselves as physical objects accessible to 2 experiments. This review concerns the studies of physical knots in condensed matter systems such as liquid crystals (LCs) and colloids [12-15] – ones that exist in three-dimensional (3D) space of these ubiquitous soft materials, that can be manipulated by laser tweezers and that can be directly observed in a microscope. Recognizing some of the key milestones in understanding the role of topology in physical behavior, the 2016 Nobel prize was awarded for theoretical discoveries of topological phase transitions and topological phases of matter [16-18], where many of the original breakthroughs resulted from considering two-dimensional (2D) systems. The situation is even more complex in 3D, where various knot-like structures can be supported, localized spatially and stabilized energetically, including both knotted filaments (vortex lines/defects/singularities) and knotted nonsingular textures such as skyrmions and hopfions [6-11]. Topological solitons are well-studied in theoretical models of high-energy physics [6,7] aiming to describe the behavior of fundamental particles and atomic nuclei. This review considers such structures in ordered soft condensed matter systems, which can be realized and characterized in detail experimentally, like in the cases of LCs and colloids. Although quantum phenomena and hard condensed matter systems received much of the recent attention in applying topology-related ideas [1], potentially even a greater playground for deploying topological concepts exists in soft condensed matter. While topological effects in soft matter encompass a much broader spectrum of phenomena [2,19,20], here we focus on knotted structures of the order parameter fields and their interaction with topologically nontrivial surfaces in LCs and colloids. Historically, knotted fields in the modern physics emerged in classical and quantum field theories [6,7,21,22] and in branches ranging from optics to chemistry, materials science, particle physics, fluid mechanics and cosmology [19,20,23-32]. Recently, knotted fields found many experimental and theoretical embodiments, including both nonsingular solitons and knotted vortices [32-48]. Knots often arise in electromagnetic fields [28,37]. For example, researchers found solutions to Maxwell’s equations with knotted and linked field lines [37]. Recent developments in optical holography and microlithography make it possible to structure the flow of light in free space whereas the rich vectorial and phase structure of sculpted light allows for different kinds of knotted and linked light beams. For example, knotted optical vortices have been embedded into laser beams by Dennis and coworkers [28]. Understanding the knots which can be embedded in holograms [28] has even led to the new classes of knots now being studied in the knot theory [8-11]. Moreover, unlike in the case of various material systems [24,26], knot structures in electromagnetic fields could be hosted in free
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